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Fractions — Complete Guide to Understanding and Solving Fractions

Grade: 6-7 | Topic: Arithmetic

What You Will Learn​

By the end of this guide you will be able to read, write, and compare fractions, perform all four arithmetic operations with them, and simplify your results to lowest terms. You will also understand how fractions connect to decimals and percentages, giving you a strong foundation for algebra and beyond.

Theory​

What is a fraction?​

A fraction represents a part of a whole. It is written as one number over another, separated by a horizontal bar:

ab\frac{a}{b}

  • Numerator (aa): the number of parts you have.
  • Denominator (bb): the total number of equal parts the whole is divided into.

For example, 38\frac{3}{8} means you have 3 out of 8 equal parts.

A fraction where the numerator is smaller than the denominator is called a proper fraction (e.g. 25\frac{2}{5}). When the numerator is equal to or greater than the denominator it is an improper fraction (e.g. 74\frac{7}{4}), which can also be written as a mixed number (1341\frac{3}{4}).

Equivalent fractions​

Two fractions are equivalent when they represent the same value. You create an equivalent fraction by multiplying or dividing both the numerator and denominator by the same non-zero number:

ab=a×kb×k(k≠0)\frac{a}{b} = \frac{a \times k}{b \times k} \quad (k \neq 0)

For instance, 12=24=36=50100\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{50}{100}. Each pair was produced by multiplying numerator and denominator by the same factor.

Simplifying fractions​

To simplify (or reduce) a fraction, divide both the numerator and denominator by their Greatest Common Divisor (GCD):

ab=a÷gcd⁡(a,b)b÷gcd⁡(a,b)\frac{a}{b} = \frac{a \div \gcd(a,b)}{b \div \gcd(a,b)}

For example, to simplify 1218\frac{12}{18}: the GCD of 12 and 18 is 6, so 1218=12÷618÷6=23\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}.

Adding and subtracting fractions​

When fractions share the same denominator, add or subtract the numerators directly:

ac+bc=a+bc\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

When the denominators differ, first rewrite each fraction with the Least Common Denominator (LCD), then combine:

ab+cd=a×d+c×bb×d\frac{a}{b} + \frac{c}{d} = \frac{a \times d + c \times b}{b \times d}

More precisely, find the LCD (the least common multiple of bb and dd), convert each fraction, then add the numerators. The general formula above works when bb and dd share no common factor; using the LCD keeps numbers smaller.

Multiplying fractions​

Multiply numerators together and denominators together:

ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}

Tip: simplify (cross-cancel) before multiplying to keep numbers manageable.

Dividing fractions​

To divide by a fraction, multiply by its reciprocal (flip the second fraction):

ab÷cd=ab×dc=a×db×c\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}

This works because division asks "how many groups of cd\frac{c}{d} fit into ab\frac{a}{b}?", and multiplying by the reciprocal answers exactly that question.

Worked Examples​

Example 1: Simplifying a fraction (easy)​

Problem: Simplify 1824\frac{18}{24} to lowest terms.

Step 1: Find the GCD of 18 and 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The greatest common factor is 6.

Step 2: Divide both numerator and denominator by 6. 1824=18÷624÷6=34\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}

Answer: 34\dfrac{3}{4}

Example 2: Adding fractions with unlike denominators (medium)​

Problem: Calculate 25+34\frac{2}{5} + \frac{3}{4}.

Step 1: Find the LCD of 5 and 4. Multiples of 5: 5, 10, 15, 20, ... Multiples of 4: 4, 8, 12, 16, 20, ... The LCD is 20.

Step 2: Rewrite each fraction with denominator 20. 25=2×45×4=820\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} 34=3×54×5=1520\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}

Step 3: Add the numerators. 820+1520=2320\frac{8}{20} + \frac{15}{20} = \frac{23}{20}

Step 4: Convert to a mixed number if desired. 2320=1320\frac{23}{20} = 1\frac{3}{20}

Answer: 2320\dfrac{23}{20} or 13201\dfrac{3}{20}

Example 3: Multiplying fractions with cross-cancellation (medium)​

Problem: Multiply 914×712\frac{9}{14} \times \frac{7}{12}.

Step 1: Before multiplying, look for common factors between any numerator and any denominator.

  • 9 and 12 share a factor of 3: simplify to 314×74\frac{3}{14} \times \frac{7}{4}.
  • 7 and 14 share a factor of 7: simplify to 32×14\frac{3}{2} \times \frac{1}{4}.

Step 2: Multiply the simplified fractions. 32×14=3×12×4=38\frac{3}{2} \times \frac{1}{4} = \frac{3 \times 1}{2 \times 4} = \frac{3}{8}

Answer: 38\dfrac{3}{8}

Example 4: Dividing fractions (medium)​

Problem: Divide 56÷109\frac{5}{6} \div \frac{10}{9}.

Step 1: Flip the second fraction to get its reciprocal. 56÷109=56×910\frac{5}{6} \div \frac{10}{9} = \frac{5}{6} \times \frac{9}{10}

Step 2: Cross-cancel before multiplying.

  • 5 and 10 share a factor of 5: simplify to 16×92\frac{1}{6} \times \frac{9}{2}.
  • 9 and 6 share a factor of 3: simplify to 12×32\frac{1}{2} \times \frac{3}{2}.

Step 3: Multiply. 12×32=34\frac{1}{2} \times \frac{3}{2} = \frac{3}{4}

Answer: 34\dfrac{3}{4}

Example 5: Mixed-number subtraction (challenging)​

Problem: Calculate 313−1343\frac{1}{3} - 1\frac{3}{4}.

Step 1: Convert each mixed number to an improper fraction. 313=3×3+13=1033\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} 134=1×4+34=741\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}

Step 2: Find the LCD of 3 and 4, which is 12. 103=4012,74=2112\frac{10}{3} = \frac{40}{12}, \quad \frac{7}{4} = \frac{21}{12}

Step 3: Subtract. 4012−2112=1912\frac{40}{12} - \frac{21}{12} = \frac{19}{12}

Step 4: Convert back to a mixed number. 1912=1712\frac{19}{12} = 1\frac{7}{12}

Answer: 17121\dfrac{7}{12}

Common Mistakes​

Mistake 1: Adding numerators and denominators directly

❌ 13+14=27\frac{1}{3} + \frac{1}{4} = \frac{2}{7}

✅ 13+14=412+312=712\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}

Why this matters: Fractions can only be added when they share the same denominator. Adding across both the top and bottom treats them like ratios being combined, which is mathematically incorrect and produces a wrong answer.

Mistake 2: Forgetting to simplify the final answer

❌ 46\frac{4}{6} left as the final answer.

✅ 46=23\frac{4}{6} = \frac{2}{3}

Why this matters: Answers in lowest terms are expected in most math courses. Always check whether the numerator and denominator share a common factor before writing your final result.

Mistake 3: Not flipping the second fraction when dividing

❌ 23÷45=2×43×5=815\frac{2}{3} \div \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}

✅ 23÷45=23×54=1012=56\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}

Why this matters: Division and multiplication are different operations. You must take the reciprocal of the divisor first, then multiply. Skipping the flip gives a completely different (and incorrect) result.

Practice Problems​

Try these on your own before checking the answers:

  1. Simplify 2436\frac{24}{36} to lowest terms.
  2. Add 56+29\frac{5}{6} + \frac{2}{9}.
  3. Multiply 815×512\frac{8}{15} \times \frac{5}{12}.
  4. Divide 78÷34\frac{7}{8} \div \frac{3}{4}.
  5. Subtract 425−2454\frac{2}{5} - 2\frac{4}{5}.
Click to see answers
  1. GCD of 24 and 36 is 12. 2436=23\frac{24}{36} = \frac{2}{3}
  2. LCD is 18. 1518+418=1918=1118\frac{15}{18} + \frac{4}{18} = \frac{19}{18} = 1\frac{1}{18}
  3. Cross-cancel (5 with 15, 8 with 12): 815×512=29\frac{8}{15} \times \frac{5}{12} = \frac{2}{9}
  4. Flip and multiply: 78×43=2824=76=116\frac{7}{8} \times \frac{4}{3} = \frac{28}{24} = \frac{7}{6} = 1\frac{1}{6}
  5. Convert: 225−145=85=135\frac{22}{5} - \frac{14}{5} = \frac{8}{5} = 1\frac{3}{5}

Summary​

  • A fraction ab\frac{a}{b} represents aa parts out of bb equal parts of a whole.
  • To add or subtract fractions, first convert them to a common denominator, then combine the numerators.
  • To multiply fractions, multiply numerators together and denominators together; cross-cancel first to keep numbers small.
  • To divide fractions, multiply by the reciprocal of the divisor (flip and multiply).
  • Always simplify your final answer by dividing numerator and denominator by their GCD.

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